Searching for the start of our infinitesimal universe

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Mathematical Speculations About the First Moment
by Bruce E. Camber Also see: Essentials and Question.

Introduction. Using simple mathematics (doublings), basic geometries, logic, and commonsense, we found 202 base-2 notations from Planck Time to this day. Within that scale, there is a domain of the infinitesimal. It appears that the smallest, first “things” of the universe are her base units, Planck Length and Planck Time. Virtually ignored for over 100 years after their formulation in 1899, these two Planck units have become de facto ISO standards that define two of the core concepts within our sciences. The largest “things” within this infinitesimal scale — particle physics and quantum fluctuations — are well-studied. Both are measurable and a key part of the corpus of today’s science. If we are ever to build consensus within the sciences, we’ve got to look at our core concepts and agree, “These are core concepts” and “this” is what each means. When we agree, we can begin to build access paths to all of our current applied sciences.

The Tetrahedron and Octahedron. Our path to these two concepts was a bit unusual. We were studying the tetrahedron in high school and we decided to explore its interior structure by dividing each edge by 2 and by connecting those new vertices. We found four smaller tetrahedrons, one in each corner, and an octahedron in the middle. We went further within all five objects.

Within that octahedron we found smaller octahedrons, one in each of its six corners, and tetrahedrons, one in each of its eight faces. We could see how we could tile and tessellate all space using both the tetrahedron and octahedron. We could also see that we could go further within each object, so we did. By our 45th time dividing by 2, we were observing lengths used within particle physics and within the measurement of quantum fluctuations. We continued to go deeper inside. Another 67 times dividing by 2, we were using numbers on the order of the Planck Length. We then used the Planck Length as our edge, and multiplied it by 2. In 112 doublings we were back in the range of our original tetrahedron. We continued. Within another 90 doublings the numbers approximated the size of the universe. At that time we learned there are 202 base-2 notations from the smallest to the largest possible measurements (a story).

STEM. We thought it was the penultimate STEM tool and began sharing it with other math teachers. Nobody had seen a base-2 chart and many pointed to similarities with Kees Boeke’s 1957 base-10 chart. Eventually we began to accept the thought that this base-2 chart just might be an original. We knew it was idiosyncratic; it was not part of any cosmological model.

Although the best scholars within the sciences have wrestled with many hypotheses about the infinitesimal scale, most of the scale can not be measured directly. The best possible approach for now is through logic and math (which, of course, includes geometry). Among the more powerful tools within this domain are natural units and the first documented work to develop natural units was by George Johnstone Stoney in 1874 in Dublin, Ireland. In 1899 Max Planck did his first calculations. In our time in 2022, the units by Planck are better known than those by Stoney.

Here are natural numbers that describe “something” within the earliest universe. Over time, we have come to believe this “something” is the starting point of this universe.

The Key Question: What does that starting point of the universe look like? Although we had natural units, we used two Platonic solids, the tetrahedron and octahedron, to tile and tessellate the universe. We had not asked if there was a particular form those Planck or Stoney natural units might take.

Circles and spheres. These are the most simple constructions — two vertices — yet not until we actually saw how tetrahedrons and octahedrons emerge from spheres were we ready to concede the role of “the starting point” to a primordial, infinitesimal sphere. The dynamic image of cubic-close packing of equal spheres found in Wikipedia opened that door. Images of the Fourier Transform pushed unexpected new frameworks for thought about spheres.

In reality, spheres are always in motion; only our images are static.

The Nature of a Sphere. To begin to know the sphere, one must know pi (π) and getting to know pi is not trivial. It may well be the oldest, best-known, most-used mathematical equation; I believe we have barely scratched its surface. A key subject within all of mathematics, and although discussed within this website, only within the first few months of 2022 have the more penetrating aspects started to become apparent.

Continuity, symmetry, and harmony. I first developed that working progression from simplicity to complexity in 1971 to describe “a perfected state within space-time.” Pi wasn’t an issue. The development of a moment of new insight was. Yet, as a result of these studies of spheres, I see that perfection is perfection and the progression from continuity to symmetry to harmony is also described by the three most basic facets of the sphere. Of course, within one’s mind’s eye, the sphere is always perfect. Those three qualities, continuity-symmetry-harmony, are deep and abiding studies within academia, however, the study of harmonic functions is its youngest, and the Fourier Transform is its science.

Wild-and-crazy Speculations. There are no less than five Wikipedia dynamic images that are being studied and speculations are being made. For example, the Fourier Transform imparts electromagnetic or gravitational qualities within each infinitesimal, primordial sphere [1]. Further, there are 539-to-4605 tredecillion spheres per second (one primordial sphere per primordial unit of time). Although such speculations are not yet recognized within the academic community, we’ve continued. Each sphere has a natural functionality of continuity-symmetry-harmony as given within pi (π) and her Fourier transform.[2] Here a finite-infinite boundary (or bridge) is defined.[3] Here each sphere manifests with its own flavors and functions which begins as an attractor or repeller.[4] As our universe expands very rapidly, those first 64 notations continue to provide unique foundations for Langlands programs, string-and-M theory, hypothetical particles, SUSY, and so much more.

Of course, these points are all idiosyncratic, unconventional, and a necessary paradigm shift. In this model gravity and electromagnetism begin to be created in the first notations. Each results from a Fourier Transform and these transforms are scale invariant.

I believe it could be the beginning of a new science and I think a good name for it is one that has been around for millennium, hypostatics, which can be loosely translated as “That which stands under” or “the foundations of the foundations.” If we start at the very beginning, within the very first moment, we can watch the universe construct itself with all the working mathematics that has already been developed. Every type of sphere could tells us something about our earliest structures. Every process will have a place. These 64 to 67 notations hold the penultimate puzzle pieces. Just maybe, this may could become an introduction to the start of a new science of the infinitesimal universe. Thank you. -BEC



[1] Each infinitesimal, primordial sphere. Somebody has to postulate it! Also, perhaps the initial conditions of “sphereness” are best described as an attractor sphere (even more fundamental than a hypothetical particle) or repeller (or repellor) sphere. Why not? It was in this discovery of the attractor scholarship that I also discovered two of its primary thinkers, Steve Smale of the University of California, Berkeley, and John Milnor of Stony Brook, Institute for Mathematical Sciences. Immediately I began to think these scholars might be able to shed light on my questions in recent emails to David Kaiser of MIT and Karen Uhlenbeck of IAS.

While engaging the work of John Milnor to define an attractor, I began thinking about the models of the Fourier Transform and those actions, although considered scale invariant, most scholars seem to hesitate to move that invariance into the infinitesimal scales. There is no hesitation here because within our base-2 chart of the universe, the infinitesimal scale of quantum fluctuations within Notation-67 quickly drops below all thresholds of direct measurements. The inclination today is to assume that Notation-1 is the manifestation of the first infinitesimal sphere and Milnor’s work describes part of the processes within that sphere.

[2] The Fourier transform. Scale invariant, these infinitesimal spheres adopt what amounts to attractor- and-repeller functionalities and these functionalities, building upon each other, manifest differently within each notation. The possibilities for complexity and uniqueness are staggering and most bewilderingly. Much more to come

[3] The finite-infinite boundary (or bridge). Pi is under-appreciated and little understood. The absolute mystery of pi is that it defines both the finite and the infinite, all within three concepts-not-3000 volumes. The inexplicable never-ending, never-repeating, always-changing, always-the-same qualities are beyond comprehension yet here can be summarized in a word, continuity, and its function is to create order. A simple formula that renders the most complex qualities is enigmatic enough, pi-and-its-spheres define space uniquely and perfectly and that is called symmetry and it is the first working relation. The majesty of that perfection is barely grasped and is profoundly unappreciated. The third quality has only been engaged for a short time in human history, perhaps a bit by Kepler, Gauss, Euler, Fourier, and Poincaré, yet new applications seem to emerge daily. Here are the dynamics of a moment, particularly the deepest, most intimate dynamics of motion that bring everything alive, is best summarized as harmony. Everything qualitative is infinite and everything quantitative is finite. A bridge of dimensionless constants connect the two. We have so much to learn and even more to begin to understand and truly appreciate. More… Even more…

[4] Spherical functionalities: From attractors and repellers, to electromagnetism and gravity. The functionalities of spheres is a young science within one of our oldest studies. Yes, the most-basic functionality of pi (π) is within the first 67 notations of the 202 that currently encapsulate the universe. Of the 67, the first ten notations are the foundations of a new science. It is well beneath the domain of quantum fluctuations as well as any direct measurement. These are the foundations under the known foundations; we’ve referred to this domain as hypostatic, for “that which stands under,” and here we take that to be continuity, symmetry, and harmony. Here are the essences, the first principles, and the starting point for every possible application and understanding of spherical functionalities.


References & Resources

Auslander, J., Bhatia, N.P., Seibert, P., Attractors in dynamical systems (PDF), NASA-CR-59858. 1964.
• Mohsen KhodadiKourosh Nozari Fazlollah HajkarimOn the viability of Planck scale cosmology with quartessenceEur. Phys. J. C 78, 716 (2018).
• John Willard Milnor (1985). “On the concept of attractor”. Communications in Mathematical Physics99 (2): 177–195. doi:10.1007/BF01212280 (excellent bibliography). Wikipedia
• Gideon Rosen, “Abstract Objects”, The Stanford Encyclopedia of Philosophy (Spring 2012 Edition), Edward N. Zalta (ed.)(Also, see Zalta, Edward, Principia Metaphysica, online PDF manuscript, 2022)
• Steve SmaleThe Emergence of Function, ArXiv, 2016
• Steve Smale, The mathematics of time Springer-Verlag, New York-Berlin, 1980. ISBN 0-387-90519-7
• David Tong, Classical Dynamics (PDF), University of Cambridge Press, Part II Mathematical Tripos, 2004, 2015
• Within this website: Transformations (2019)

Cubic-close packing of equal spheres: Though lifted up within this website in 2016, this image naturally builds on the next five dynamic images.
• Fourier-A: Sine-Cosine-Waves: We’ll all learn about the wave’s fundamental relationship to the circle.
Fourier-B: Focus on Sine: We’re all going to learn a little trigonometry and calculus, too.
• Possible Fourier-C-D: An all-natural polarization: An open metaphor and analogical construction.
Possible Fourier-F: Lagrange points: Lagrangians come home.



30 April 2022: Willy Fischler
30 April 2022: Richard Streit Hamilton
25 April 2022: Indika Rajapakse
24 April 2022: John Milnor
23 April 2022: Steve Smale
21 April 2022, Silvia Milana, Nature, London
20 April 2022 @ 11:38 AM Rashmi Shivni



2:07 PM · May 5, 2022 @BillGates, @sundarpichai (Apple CEO) The best cosmologists-physicists, people like Princeton’s James Peebles, Nobel Laureate 2019, say we do not have a theory for the first microseconds of the universe. Perhaps your most brilliant people can work on it: might be helpful place to start.

2:17 PM · May 1, 2022 @FinitePhysicist Can the finite be quantitative and the infinite be qualitative? Does pi (π) (and how it never ends) tell us about continuity, a perfect sphere about symmetry, and its harmonic functions about dynamics?

2:06 PM · May 1, 2022 @Pontifex Teach us to grasp the infinite. Rediscover pi (π) and how it never ends (continuity) and its symmetry is perfect, and its harmonic functions pervasive. There’s the infinite, all qualitative, and the finite is quantitative: You can do it. We can!

1:15 PM · Apr 26, 2022 @CERNCourier On July 10, 2021 your tweet was about the next generation of detector designs. What about the Planck scale? It is so far beyond detectors, a new approach is needed. The first 64 notations of 202 base-2 that contain the universe need study:

8:05 AM · Apr 21, 2022 @TheDailyShow We all do circular arguments. We’re locked up within little worldviews when a view of the entire universe is needed. How presumptuous it is to think the universe is us when our solar system is part of the Milky Way and… Here’s the universe:

4:08 PM · Apr 9, 2022. @anabelquanhaase We’re caught by three historic errors, one by Aristotle: another by Newton, and also by Hawking as understood by his co-author Neil Turok: We’re confused because our grasp of the foundations is wrong. Also, see:

Looking back…


Keys to this document, hypostatics

  • This page was started on Tuesday, April 5, 2022.
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  • The second headline is: Mathematical Speculations About the Very Beginning
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  • The first byline is: Introducing the start of a new science of the infinitesimal universe
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